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Uspekhi Mat. Nauk, 2010, Volume 65, Issue 1(391), Pages 145–176 (Mi umn9345)  

This article is cited in 5 scientific papers (total in 5 papers)

Closed 1-forms in topology and geometric group theory

M. Farbera, R. Geogheganb, D. Schütza

a University of Durham, UK
b State University of New York, Binghamton, USA

Abstract: In this article we describe relations of the topology of closed 1-forms to the group-theoretic invariants of Bieri–Neumann–Strebel–Renz. Starting with a survey, we extend these Sigma invariants to finite CW-complexes and show that many properties of the group-theoretic version have analogous statements. In particular, we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik–Schnirelmann category of a closed 1-form and to the existence of a non-singular closed 1-form in a given cohomology class on a high-dimensional closed manifold.
Bibliography: 32 titles.

Keywords: Sigma invariants, Lusternik–Schnirelmann category, Novikov ring, movability of homology classes.

DOI: https://doi.org/10.4213/rm9345

Full text: PDF file (814 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2010, 65:1, 143–172

Bibliographic databases:

UDC: 515.165
MSC: Primary 20J05, 58E05; Secondary 55M30
Received: 06.10.2008

Citation: M. Farber, R. Geoghegan, D. Schütz, “Closed 1-forms in topology and geometric group theory”, Uspekhi Mat. Nauk, 65:1(391) (2010), 145–176; Russian Math. Surveys, 65:1 (2010), 143–172

Citation in format AMSBIB
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\by M.~Farber, R.~Geoghegan, D.~Sch\"utz
\paper Closed 1-forms in topology and geometric group theory
\jour Uspekhi Mat. Nauk
\yr 2010
\vol 65
\issue 1(391)
\pages 145--176
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\jour Russian Math. Surveys
\yr 2010
\vol 65
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\pages 143--172
\crossref{https://doi.org/10.1070/RM2010v065n01ABEH004663}
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    This publication is cited in the following articles:
    1. Suciu A.I., “Fundamental groups, Alexander invariants, and cohomology jumping loci”, Topology of algebraic varieties and singularities, Contemp. Math., 538, Amer. Math. Soc., Providence, RI, 2011, 179–223  crossref  mathscinet  zmath  isi
    2. T. Hüttemann, D. Quinn, “Finite domination and Novikov rings: Laurent polynomial rings in two variables”, J. Algebra Appl., 14:4 (2015), 1550055, 44 pp.  crossref  mathscinet  zmath  isi
    3. Dowdall S., Kapovich I., Leininger Ch.J., “Dynamics on Free-By-Cyclic Groups”, 19, no. 5, 2015, 2801–2899  crossref  mathscinet  zmath  isi
    4. Friedl S., Maxim L., “Twisted Novikov homology of complex hypersurface complements”, Math. Nachr., 290:4 (2017), 604–612  crossref  mathscinet  zmath  isi  scopus
    5. Dowdall S., Kapovich I., Leininger Ch.J., “Mcmullen Polynomials and Lipschitz Flows For Free-By-Cyclic Groups”, J. Eur. Math. Soc., 19:11 (2017), 3253–3353  crossref  mathscinet  zmath  isi
  • Успехи математических наук Russian Mathematical Surveys
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